Y=-2x^2-20x-48.
Does the relation have a maximum or minimum value?
Identify the y-intercept.
Factor the equation.
Identify x-intercepts of the relation.
What is the equation of the axis of symmetry?
Find the coordinates of the vertex.
The relation has a maximum value because the coefficient of the x^2 term is negative.
The y-intercept is the value of y when x=0:
Y = -2(0)^2 - 20(0) - 48
Y = -48
So, the y-intercept is (0, -48).
To factor the equation, we can first factor out a common factor of -2:
Y = -2(x^2 + 10x + 24)
Y = -2(x + 6)(x + 4)
To find the x-intercepts, we set y=0 and solve for x:
0 = -2x^2 - 20x - 48
0 = x^2 + 10x + 24
0 = (x + 6)(x + 4)
So, the x-intercepts are (-6, 0) and (-4, 0).
The equation of the axis of symmetry can be found using the formula x = -b/2a:
x = -(-20)/(2*-2)
x = 10/(-4)
x = -2.5
So, the equation of the axis of symmetry is x = -2.5.
To find the coordinates of the vertex, we substitute x=-2.5 into the equation:
Y = -2(-2.5)^2 - 20(-2.5) - 48
Y = -2(6.25) + 50 - 48
Y = -12.5 + 50 - 48
Y = -10.5
Therefore, the coordinates of the vertex are (-2.5, -10.5).